Instead of a cordless iron could you rig up a cord minder? I,ve seen some for sale that attach to the ironing board to keep the cord out of the way and they look like they may solve the problem without the drawbacks of a cordless iron (cooling down before finishing).
my ironing station didn't have a place to put one of the commercially available cord minders so I rigged up one for myself using some copper wire to suspend an old wooden thread spool from the shelf next to (and higher than) my ironing station. The cord goes over the wooden spool and I tied a fishing weight and some metal washers to it to serve as a counter weight. Now my cord no longer drags across my ironing surface and it stays tangle free.
Rob